Reckoning / Shape
Room · 42 figures and solids
Shape
Areas, perimeters, volumes and surfaces. Change the dimensions and watch the figure and the numbers follow. Every formula is checked at build time against an independent measurement of the shape.
The topology lens: which of these shapes are “the same” if you may stretch but not tear.
Plane figures
Quadrilaterals
Square
Four equal sides and four right angles.
Rectangle
Opposite sides equal and parallel, four right angles.
Parallelogram
Two pairs of parallel sides. The height is measured at right angles to the base, not along the slanted side.
Rhombus
Four equal sides; the diagonals cross at right angles and halve each other.
Trapezoid
Exactly one pair of parallel sides. Careful: American and British usage swap the names trapezoid and trapezium.
Any quadrilateral
Four straight sides, none necessarily parallel. Any convex quadrilateral can be measured from its two diagonals and the angle between them.
Plane figures
Triangles
Triangle from three sides
Any triangle. With only the three sides known, the area comes from Heron’s formula.
Right triangle
One angle is 90°. The side opposite it, the hypotenuse, is the longest.
Isosceles triangle
Two equal sides; the height from the apex halves the base at right angles.
Equilateral triangle
Three equal sides and three 60° angles.
Plane figures
Circles
Circle
All points at the same distance r from the centre.
Circular sector
A slice of a disc between two radii.
Circular segment
The part of a disc cut off by a chord.
Annulus (ring)
The flat ring between two circles with the same centre, like a washer.
Ring sector
A slice of an annulus, like a curved tile or a voussoir of an arch.
Ellipse
A circle stretched in one direction: the set of points whose distances to two foci add to 2a.
Spandrel (fillet)
The corner left between a right angle and a quarter circle that touches both sides, as in a rounded corner or a weld fillet.
Plane figures
Polygons
Solids
Prisms and pyramids
Cube
Six equal square faces.
Rectangular box (cuboid)
Six rectangular faces meeting at right angles.
Regular prism
Two equal regular polygons joined by rectangles.
Square pyramid
A square base with four triangles meeting at the apex above its centre.
Truncated pyramid (frustum)
A square pyramid with its top cut off parallel to the base.
Wedge
A rectangular base a by b and a top ridge of length c parallel to a, at height h.
Prismoid
Two parallel rectangular ends of different sizes joined by flat faces, as in a cutting, embankment or hopper.
Solids
Round solids
Cylinder
Two equal parallel circles joined by a curved side.
Hollow cylinder (tube)
A cylinder with a coaxial cylinder removed, like a pipe.
Obliquely cut cylinder
A cylinder cut by a slanted plane, like a ventilation duct elbow piece.
Cone
A circular base and a curved side narrowing to a point above its centre.
Truncated cone (frustum)
A cone with its top cut off parallel to the base, like a bucket or a lampshade.
Solids
Spheres
Sphere
All points at distance r from the centre: the solid ball and its surface.
Spherical cap (segment of one base)
The piece of a ball cut off by one plane, like a dome.
Spherical zone (segment of two bases)
The slice of a ball between two parallel planes.
Spherical sector
A cone with its apex at the centre of a ball, capped by the sphere: an ice-cream cone shape.
Spherical wedge
The slice of a ball between two half-planes through its axis, like a segment of an orange.
Hollow sphere (shell)
A ball with a concentric ball removed, like a float or a tennis ball.
Ellipsoid
A sphere stretched along three axes by different amounts. With two axes equal it is a spheroid, the shape of the Earth.
Solids
Rings
Torus (anchor ring)
A circle of radius r swept round an axis at distance R: a ring, a doughnut, an O-ring.
Ring of square section
A square of side w swept round an axis at distance R from its centre, like a flat washer or a gasket.
Barrel
A solid with circular ends and bulging sides. Here the staves are parabolic arcs.